<s>
In	O
mathematical	O
logic	O
,	O
basic	B-General_Concept
fuzzy	I-General_Concept
logic	I-General_Concept
(	O
or	O
shortly	O
BL	B-General_Concept
)	O
,	O
the	O
logic	O
of	O
the	O
continuous	O
t-norms	B-General_Concept
,	O
is	O
one	O
of	O
the	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
.	O
</s>
<s>
It	O
belongs	O
to	O
the	O
broader	O
class	O
of	O
substructural	O
logics	O
,	O
or	O
logics	O
of	O
residuated	O
lattices	O
;	O
it	O
extends	O
the	O
logic	O
MTL	B-General_Concept
of	O
all	O
left-continuous	O
t-norms	B-General_Concept
.	O
</s>
<s>
The	O
language	O
of	O
the	O
propositional	O
logic	O
BL	B-General_Concept
consists	O
of	O
countably	O
many	O
propositional	O
variables	O
and	O
the	O
following	O
primitive	O
logical	O
connectives	O
:	O
</s>
<s>
Bottom	O
(	O
nullary	O
—	O
a	O
propositional	O
constant	O
)	O
;	O
or	O
are	O
common	O
alternative	O
signs	O
and	O
zero	O
a	O
common	O
alternative	O
name	O
for	O
the	O
propositional	O
constant	O
(	O
as	O
the	O
constants	O
bottom	O
and	O
zero	O
of	O
substructural	O
logics	O
coincide	O
in	O
MTL	B-General_Concept
)	O
.	O
</s>
<s>
Well-formed	O
formulae	O
of	O
BL	B-General_Concept
are	O
defined	O
as	O
usual	O
in	O
propositional	O
logics	O
.	O
</s>
<s>
A	O
Hilbert-style	O
deduction	O
system	O
for	O
BL	B-General_Concept
has	O
been	O
introduced	O
by	O
Petr	O
Hájek	O
(	O
1998	O
)	O
.	O
</s>
<s>
Like	O
in	O
other	O
propositional	O
t-norm	B-General_Concept
fuzzy	I-General_Concept
logics	I-General_Concept
,	O
algebraic	O
semantics	O
is	O
predominantly	O
used	O
for	O
BL	B-General_Concept
,	O
with	O
three	O
main	O
classes	O
of	O
algebras	O
with	O
respect	O
to	O
which	O
the	O
logic	O
is	O
complete	O
:	O
</s>
<s>
Standard	O
semantics	O
,	O
formed	O
of	O
all	O
standard	O
BL-algebras	O
—	O
that	O
is	O
,	O
all	O
BL-algebras	O
whose	O
lattice	O
reduct	O
is	O
the	O
real	O
unit	O
interval	O
 [ 0 , 1 ] 	O
with	O
the	O
usual	O
order	O
;	O
they	O
are	O
uniquely	O
determined	O
by	O
the	O
function	O
that	O
interprets	O
strong	O
conjunction	O
,	O
which	O
can	O
be	O
any	O
continuous	O
t-norm	B-General_Concept
.	O
</s>
